Antimagic Labelings of Regular Bipartite Graphs Daniel Cranston - - PowerPoint PPT Presentation

Antimagic Labelings of Regular Bipartite Graphs

Daniel Cranston dcransto@dimacs.rutgers.edu DIMACS, Rutgers University

Antimagic Labelings

• Def. magic labeling:

an injection from the edges of G to {1, 2, . . . , |E|} such that the sum of the labels incident to each ver- tex is the same

• Def. a graph is magic if it has an magic labeling

Antimagic Labelings

• Def. antimagic labeling:

an injection from the edges of G to {1, 2, . . . , |E|} such that the sum of the labels incident to each ver- tex is distinct

• Def. a graph is antimagic if it has an antimagic labeling

Antimagic Labelings

• Def. antimagic labeling:

an injection from the edges of G to {1, 2, . . . , |E|} such that the sum of the labels incident to each ver- tex is distinct

• Def. a graph is antimagic if it has an antimagic labeling

Conj. [Ringel 1990] Every connected graph other than K2 is antimagic.

Antimagic Labelings

• Def. antimagic labeling:

an injection from the edges of G to {1, 2, . . . , |E|} such that the sum of the labels incident to each ver- tex is distinct

• Def. a graph is antimagic if it has an antimagic labeling

Conj. [Ringel 1990] Every connected graph other than K2 is antimagic. Thm. [Alon et al. 2004] ∃C s.t. ∀ n if G has n vertices and δ(G) ≥ C log n, then G is antimagic.

Antimagic Labelings

• Def. antimagic labeling:

an injection from the edges of G to {1, 2, . . . , |E|} such that the sum of the labels incident to each ver- tex is distinct

• Def. a graph is antimagic if it has an antimagic labeling

Conj. [Ringel 1990] Every connected graph other than K2 is antimagic. Thm. [Alon et al. 2004] ∃C s.t. ∀ n if G has n vertices and δ(G) ≥ C log n, then G is antimagic. Thm. [Alon et al. 2004] If ∆(G) ≥ n − 2, then G is antimagic.

Antimagic Labelings

• Def. antimagic labeling:

an injection from the edges of G to {1, 2, . . . , |E|} such that the sum of the labels incident to each ver- tex is distinct

• Def. a graph is antimagic if it has an antimagic labeling

Conj. [Ringel 1990] Every connected graph other than K2 is antimagic. Thm. [Alon et al. 2004] ∃C s.t. ∀ n if G has n vertices and δ(G) ≥ C log n, then G is antimagic. Thm. [Alon et al. 2004] If ∆(G) ≥ n − 2, then G is antimagic.

• Pf. for ∆(G) = n − 1. Let d(v) = n − 1. Label G − v arbitrarily.

Label the final star in order of partial sum.

Antimagic Labelings

• Def. antimagic labeling:

an injection from the edges of G to {1, 2, . . . , |E|} such that the sum of the labels incident to each ver- tex is distinct

• Def. a graph is antimagic if it has an antimagic labeling

Conj. [Ringel 1990] Every connected graph other than K2 is antimagic. Thm. [Alon et al. 2004] ∃C s.t. ∀ n if G has n vertices and δ(G) ≥ C log n, then G is antimagic. Thm. [Alon et al. 2004] If ∆(G) ≥ n − 2, then G is antimagic.

• Pf. for ∆(G) = n − 1. Let d(v) = n − 1. Label G − v arbitrarily.

Label the final star in order of partial sum. Thm. [Alon et al. 2004] Every complete partite graph other than K2 is antimagic.

Antimagic Labelings

Thm. [Cranston 2007] All k-regular bipartite graphs with k ≥ 2 are antimagic.

Antimagic Labelings

Thm. [Cranston 2007] All k-regular bipartite graphs with k ≥ 2 are antimagic.

• Prop. Cycles are antimagic.

Antimagic Labelings

Thm. [Cranston 2007] All k-regular bipartite graphs with k ≥ 2 are antimagic.

• Prop. Cycles are antimagic.

Pf. 1 3 5 n − 3 n − 1 2 4 6 n − 2 n

Antimagic Labelings

Thm. [Cranston 2007] All k-regular bipartite graphs with k ≥ 2 are antimagic.

• Prop. Cycles are antimagic.

Pf. 1 3 5 n − 3 n − 1 2 4 6 n − 2 n

• Prop. If G1 and G2 are k-regular and antimagic, then so is their

disjoint union.

Antimagic Labelings

Thm. [Cranston 2007] All k-regular bipartite graphs with k ≥ 2 are antimagic.

• Prop. Cycles are antimagic.

Pf. 1 3 5 n − 3 n − 1 2 4 6 n − 2 n

• Prop. If G1 and G2 are k-regular and antimagic, then so is their

disjoint union.

• Pf. Increase each label on G2 by m1.

Antimagic Labelings

Thm. [Cranston 2007] All k-regular bipartite graphs with k ≥ 2 are antimagic.

• Prop. Cycles are antimagic.

Pf. 1 3 5 n − 3 n − 1 2 4 6 n − 2 n

• Prop. If G1 and G2 are k-regular and antimagic, then so is their

disjoint union.

• Pf. Increase each label on G2 by m1.

Today, I’ll prove the theorem for k ≥ 5 odd.

Odd degree at least 5

constructing an antimagic labeling: resolving all potential conflicts

Odd degree at least 5

constructing an antimagic labeling: resolving all potential conflicts Plan for odd degree 2l + 5

◮ Decompose G into regular subgraphs G1, G2.

Odd degree at least 5

constructing an antimagic labeling: resolving all potential conflicts Plan for odd degree 2l + 5

◮ Decompose G into regular subgraphs G1, G2. ◮ G1 is (2l + 2)-regular and resolves conflicts between A and B.

Odd degree at least 5

constructing an antimagic labeling: resolving all potential conflicts Plan for odd degree 2l + 5

◮ Decompose G into regular subgraphs G1, G2. ◮ G1 is (2l + 2)-regular and resolves conflicts between A and B. ◮ in G1, sums in A equal t and sums in B not equal t(mod3)

Odd degree at least 5

constructing an antimagic labeling: resolving all potential conflicts Plan for odd degree 2l + 5

◮ Decompose G into regular subgraphs G1, G2. ◮ G1 is (2l + 2)-regular and resolves conflicts between A and B. ◮ in G1, sums in A equal t and sums in B not equal t(mod3) ◮ G2 is 3-regular and resolves conflicts within A and within B.

Odd degree at least 5

constructing an antimagic labeling: resolving all potential conflicts Plan for odd degree 2l + 5

◮ Decompose G into regular subgraphs G1, G2. ◮ G1 is (2l + 2)-regular and resolves conflicts between A and B. ◮ in G1, sums in A equal t and sums in B not equal t(mod3) ◮ G2 is 3-regular and resolves conflicts within A and within B. ◮ in G2, sums in A and in B are distinct multiples of 3

Odd degree at least 5

constructing an antimagic labeling: resolving all potential conflicts Plan for odd degree 2l + 5

◮ Decompose G into regular subgraphs G1, G2. ◮ G1 is (2l + 2)-regular and resolves conflicts between A and B. ◮ in G1, sums in A equal t and sums in B not equal t(mod3) ◮ G2 is 3-regular and resolves conflicts within A and within B. ◮ in G2, sums in A and in B are distinct multiples of 3 ◮ order sums in G2 in B to match order of sums in G1 in B

Odd degree at least 5

constructing an antimagic labeling: resolving all potential conflicts Plan for odd degree 2l + 5

◮ Decompose G into regular subgraphs G1, G2. ◮ G1 is (2l + 2)-regular and resolves conflicts between A and B. ◮ in G1, sums in A equal t and sums in B not equal t(mod3) ◮ G2 is 3-regular and resolves conflicts within A and within B. ◮ in G2, sums in A and in B are distinct multiples of 3 ◮ order sums in G2 in B to match order of sums in G1 in B

42 42 42 42 16 41 41 70 G1

Odd degree at least 5

constructing an antimagic labeling: resolving all potential conflicts Plan for odd degree 2l + 5

◮ Decompose G into regular subgraphs G1, G2. ◮ G1 is (2l + 2)-regular and resolves conflicts between A and B. ◮ in G1, sums in A equal t and sums in B not equal t(mod3) ◮ G2 is 3-regular and resolves conflicts within A and within B. ◮ in G2, sums in A and in B are distinct multiples of 3 ◮ order sums in G2 in B to match order of sums in G1 in B

24+42 30+42 21+42 27+42 21+16 24+41 27+41 30+70 G1 ∪ G2

Odd degree at least 5

Lem. Let G be bipartite of degree 2l + 2. Let t = (l + 1)(2ln + 1). We can label G so the sum at each vertex of A is t and at each vertex of B is not equal to t(mod3).

Odd degree at least 5

Lem. Let G be bipartite of degree 2l + 2. Let t = (l + 1)(2ln + 1). We can label G so the sum at each vertex of A is t and at each vertex of B is not equal to t(mod3).

• Pf. Partition labels into pairs with sum 2ln + 1: (0, 0) and (1, 2)

mod 3. Decompose 2l-factor into l 2-factors; at each vertex of A use pair of labels, at each vertex of B labels sum to 0(mod3). Remaining 2-factor: at each vertex of A use pair of labels, at each vertex of B labels don’t sum to 0(mod3).

Odd degree at least 5

Lem. Let G be bipartite of degree 2l + 2. Let t = (l + 1)(2ln + 1). We can label G so the sum at each vertex of A is t and at each vertex of B is not equal to t(mod3).

• Pf. Partition labels into pairs with sum 2ln + 1: (0, 0) and (1, 2)

mod 3. Decompose 2l-factor into l 2-factors; at each vertex of A use pair of labels, at each vertex of B labels sum to 0(mod3). Remaining 2-factor: at each vertex of A use pair of labels, at each vertex of B labels don’t sum to 0(mod3). l 2-factors like this

Odd degree at least 5

Lem. Let G be bipartite of degree 2l + 2. Let t = (l + 1)(2ln + 1). We can label G so the sum at each vertex of A is t and at each vertex of B is not equal to t(mod3).

• Pf. Partition labels into pairs with sum 2ln + 1: (0, 0) and (1, 2)

mod 3. Decompose 2l-factor into l 2-factors; at each vertex of A use pair of labels, at each vertex of B labels sum to 0(mod3). Remaining 2-factor: at each vertex of A use pair of labels, at each vertex of B labels don’t sum to 0(mod3). l 2-factors like this

Odd degree at least 5

Lem. Let G be bipartite of degree 2l + 2. Let t = (l + 1)(2ln + 1). We can label G so the sum at each vertex of A is t and at each vertex of B is not equal to t(mod3).

• Pf. Partition labels into pairs with sum 2ln + 1: (0, 0) and (1, 2)

mod 3. Decompose 2l-factor into l 2-factors; at each vertex of A use pair of labels, at each vertex of B labels sum to 0(mod3). Remaining 2-factor: at each vertex of A use pair of labels, at each vertex of B labels don’t sum to 0(mod3). 1 1 1 1 2 2 2 2 l 2-factors like this

Odd degree at least 5

Lem. Let G be bipartite of degree 2l + 2. Let t = (l + 1)(2ln + 1). We can label G so the sum at each vertex of A is t and at each vertex of B is not equal to t(mod3).

• Pf. Partition labels into pairs with sum 2ln + 1: (0, 0) and (1, 2)

mod 3. Decompose 2l-factor into l 2-factors; at each vertex of A use pair of labels, at each vertex of B labels sum to 0(mod3). Remaining 2-factor: at each vertex of A use pair of labels, at each vertex of B labels don’t sum to 0(mod3). 1 1 1 1 2 2 2 2 l 2-factors like this One 2-factor like this

Odd degree at least 5

Lem. Let G be bipartite of degree 2l + 2. Let t = (l + 1)(2ln + 1). We can label G so the sum at each vertex of A is t and at each vertex of B is not equal to t(mod3).

• Pf. Partition labels into pairs with sum 2ln + 1: (0, 0) and (1, 2)

mod 3. Decompose 2l-factor into l 2-factors; at each vertex of A use pair of labels, at each vertex of B labels sum to 0(mod3). Remaining 2-factor: at each vertex of A use pair of labels, at each vertex of B labels don’t sum to 0(mod3). 1 1 1 1 2 2 2 2 l 2-factors like this 1 2 1 2 1 2 One 2-factor like this

Odd degree at least 5

Lem. Let G be a 3-regular bipartite graph with parts A and B. Let bi be an ordering of the vertices of B. We can label G with the integers 1 through 3n so that at each bi the sum is 3n + 3i and for each i exactly one vertex in A has sum 3n + 3i.

Odd degree at least 5

Lem. Let G be a 3-regular bipartite graph with parts A and B. Let bi be an ordering of the vertices of B. We can label G with the integers 1 through 3n so that at each bi the sum is 3n + 3i and for each i exactly one vertex in A has sum 3n + 3i.

• Pf. Decompose G into three 1-factors

Odd degree at least 5

Lem. Let G be a 3-regular bipartite graph with parts A and B. Let bi be an ordering of the vertices of B. We can label G with the integers 1 through 3n so that at each bi the sum is 3n + 3i and for each i exactly one vertex in A has sum 3n + 3i.

• Pf. Decompose G into three 1-factors

bi ai 3i − 2 3 n + 3

• 3

l 3i-1

Odd degree at least 5

Lem. Let G be a 3-regular bipartite graph with parts A and B. Let bi be an ordering of the vertices of B. We can label G with the integers 1 through 3n so that at each bi the sum is 3n + 3i and for each i exactly one vertex in A has sum 3n + 3i.

• Pf. Decompose G into three 1-factors

bi ai 3i − 2 3 n + 3 − 3 i 3i − 1

Odd degree at least 5

Lem. Let G be a 3-regular bipartite graph with parts A and B. Let bi be an ordering of the vertices of B. We can label G with the integers 1 through 3n so that at each bi the sum is 3n + 3i and for each i exactly one vertex in A has sum 3n + 3i.

• Pf. Decompose G into three 1-factors

bi ai 3i − 2 3 n + 3 − 3 i 3i − 1 at bi sum is (3i − 2) + (3n + 3 − 3j) + (3j − 1) = 3n + 3i

Odd degree at least 5

Lem. Let G be a 3-regular bipartite graph with parts A and B. Let bi be an ordering of the vertices of B. We can label G with the integers 1 through 3n so that at each bi the sum is 3n + 3i and for each i exactly one vertex in A has sum 3n + 3i.

• Pf. Decompose G into three 1-factors

bi ai 3i − 2 3 n + 3 − 3 i 3i − 1 at bi sum is (3i − 2) + (3n + 3 − 3j) + (3j − 1) = 3n + 3i at ai sum is (3i − 1) + (3n + 3 − 3l) + (3l − 2) = 3n + 3i

Odd degree at least 5

Lem. Let G be a 3-regular bipartite graph with parts A and B. Let bi be an ordering of the vertices of B. We can label G with the integers 1 through 3n so that at each bi the sum is 3n + 3i and for each i exactly one vertex in A has sum 3n + 3i.

• Pf. Decompose G into three 1-factors

bi ai 3i − 2 3 n + 3 − 3 i 3i − 1 at bi sum is (3i − 2) + (3n + 3 − 3j) + (3j − 1) = 3n + 3i at ai sum is (3i − 1) + (3n + 3 − 3l) + (3l − 2) = 3n + 3i Hence, every regular bipartite graph of odd degree ≥ 5 is antimagic.

Thank you!